Questions: Find the coordinates of the missing endpoint if B is the midpoint of AC. A(2,1), B(10,4) C

Solution Steps

To find the coordinates of the missing endpoint \( C \) when \( B \) is the midpoint of \( \overline{AC} \), we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint \( B \) are the averages of the coordinates of \( A \) and \( C \). Given \( A(x_1, y_1) \) and \( B(x_2, y_2) \), we can set up the following equations to solve for the coordinates of \( C(x_3, y_3) \):

\[ x_2 = \frac{x_1 + x_3}{2} \] \[ y_2 = \frac{y_1 + y_3}{2} \]

We can solve these equations to find \( x_3 \) and \( y_3 \).

Given: \( A(2, 1) \) \( B(10, 4) \)

We need to find the coordinates of \( C(x_3, y_3) \) such that \( B \) is the midpoint of \( \overline{AC} \).

The midpoint formula states: \[ B\left(\frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}\right) \]

Given \( B(10, 4) \), we can set up the following equations: \[ 10 = \frac{2 + x_3}{2} \] \[ 4 = \frac{1 + y_3}{2} \]

Solving the first equation for \( x_3 \): \[ 10 = \frac{2 + x_3}{2} \implies 20 = 2 + x_3 \implies x_3 = 18 \]

Solving the second equation for \( y_3 \): \[ 4 = \frac{1 + y_3}{2} \implies 8 = 1 + y_3 \implies y_3 = 7 \]

Final Answer